Procedimientos

Heat maps of the percent error for each combination of excitation angle and TR for the closed system are shown in Figure 4-2. A wide range of excitation angle and TR combinations resulted in accurate measurement of exchange rates, nominally with errors less than about 10% of the driving exchange rate. The range of sequence parameters that resulted in accurate fits was larger when a higher apparent exchange rate was used in the driving model. Estimates of the driving exchange rate began to result in inaccurate rate constants at very low and relatively high excitation angles, with a weaker dependence on TR. Accurate fit exchange rates were achieved with excitation angles of 10° to 40° for nearly all TRs at both high and low simulation exchange rates. Notably, the accuracy of analysis degrades precipitously for combinations with low exchange and large excitation angles as excitation losses suppressed the entire hyperpolarized signal before a significant lactate signal could be produced.

Figure 2-2. Percent error plots of driving versus fit exchange rates for the closed system approximation. Left, high simulation exchange rate of 0.1 s-1_{; right, low simulation exchange rate of 0.02 s}-1_{. Errors}

ranged from 1% to greater than 250%. A wide range of sequence parameters provided accurate estimations of 𝑘̂ , especially for the high simulation exchange rate data. 𝑝𝑙

When perfusion was included, the accuracy of these measurements (Figure 4-3) at the lower driving exchange rate did not deteriorate to the same extent as was seen in the closed system. Generally, a more limited range of sequence parameter combinations yielded accurate observations though the maximum overall error was reduced. Regarding the high driving exchange rate, accuracy of measurements begins to degrade along a boundary extending approximately from an excitation angle of 30° and TR of 2 seconds to an excitation angle of 70° and TR of 10 s. Data assuming a lower driving exchange rate resulted in substantial error (~30% or greater) except over a narrow band from excitation angle 20o _{and TR of 2 to excitation angle 30}o _{and TR of 10 as the limited lactate signal produced by slow}

exchange was more sensitive to the effects of excitation on signal evolution.

Figure 4-3. Percent error plots of driving versus fit exchange rates for the perfused system

of 0.02 s-1_{. The errors ranged from 1% to over 200%. Generally, the errors were less drastic than those}

for the closed system. However, there were fewer combinations of sequence parameters that yielded highly accurate exchange rate estimations.

Total SNR and average SNR per excitation were used as metrics of signal quality. The effects of excitation angle and TR on these metrics are summarized in Figures 4-4 and 4-5. The total SNR is

maximized at fast repetition times and relatively low excitation angles for the closed system (Figure 4-4). In contrast to the closed system, a wider range of excitation angles resulted in maximal total SNR for the perfused system likely due to vascular delivery of fresh pyruvate offsetting the signal losses at higher excitation angles. The average SNR per excitation, in contrast, peaks at higher excitation angles with longer TRs (Figure 4-5). It is important to note that the sequence parameter combinations that result in very low total SNR (Figure 4-4) or average SNR per excitation (Figure 4-5) do not correspond well to regions of high fit error (Figures 4-2 and 4-3) except at the lowest excitation angles.

Figure 4-4. Relative total SNR of each study for the high driving exchange rate for both closed and perfused system. The total SNR peaks at a moderate excitation angle and short repetition time for the

closed system as opposed to the perfused system where the total SNR is relatively independent of excitation angle except at the lowest excitation angles. The results are similar for lower driving exchange (data not shown).

Figure 4-5. Average SNR per excitation for the closed and perfused systems with a high driving exchange rate. The average SNR is greater at higher excitation angles and longer TRs. Additionally, the SNR of the perfused system has a weak dependence on TR for higher excitation angles. Average SNR plots are similar for lower driving exchange (data not shown).

To explore the cause of fitting errors, we considered fit residual as a metric of fitting

performance. The normalized square-2 norm of the fits for the high conversion rate (Figures. 4-2a, 4-3a) are shown in Figure 4-6. For the closed system, the norm increases with larger excitation angles with a slight dependence on TR. In contrast, the perfused system shows fairly low and uniform residuals. Higher fit norms (Figure 4-6) do not correlate with parameter combinations that resulted in inaccurate fitting of the exchange rate (Figures 4-2a, 4-3a), which implies that fit quality alone cannot explain inaccurate fitting results.

Figure 4-6. Normalized Square-2 norms of the fits for both closed and perfused systems, with a high driving kpl. The norms for the closed system rise rapidly at higher excitation angles. In contrast, the

norms for the perfused system are uniformly lower. The norms of both closed and perfused systems have limited dependence on TR. Similar results were observed with a lower driving exchange (data not shown).

Because of a fundamental motivating interest in the detection of changes in metabolism by MRS of hyperpolarized (HP)-pyruvate, we sought to determine which set of sequence parameters would provide the most accurate measurement of differences between high and low driving exchange rates. Maps for the error in the observed differences, or contrast error, for the closed and perfused systems are shown in Figure 4-7. In general, regions of sequence parameter values that result in the most accurate measurement of contrast closely match the corresponding regions for data reflecting the higher driving exchange rates. This is not true at the highest excitation angles, where very large errors in analysis of low driving exchange rate more significantly affect the differences that were observed.

Figure 4-7. Contrast error maps for the closed (left) and perfused (right) system approximations. The errors ranged from 1% to more than 100%. The large discrepancy between the simulation exchange rates in the two systems led to accuracy plots that closely matched the higher exchange rate plots.

Using instantaneous excitation loss modeling in equation (4.6), figures 4-2 and 4-3 were

recalculated. The results in figures 4-8 and 4-9 show a slight but meaningful divergence from the results modeled with excitation losses averaged over the entire repetition time. Generally, fitting with the instantaneous excitation losses allowed for more accurate 𝑘𝑝𝑙 measurement at longer repetition times

and larger excitation angles and had little effect on accuracy for smaller excitation angles. This illustrates the limitations of the averaged excitation loss model and highlights the importance of calculating the basic physics modeled by the Bloch simulator. The errors introduced by the modeling assumptions, i.e., the differences between 4-8 and 4-9 vs 4-2 and 4-3, can be found by processing the same simulation data with different modeling assumptions.

Figure 4-8. Percent error plots of driving versus fit exchange rates for the closed system approximation. These are similar to Figure 4-2 but are fit with equation (4.6). Left, high simulation exchange rate of 0.1 s-1_{; right, low simulation exchange rate of 0.02 s}-1_.

Figure 4-9. Percent error plots of driving versus fit exchange rates for the perfused system. These are similar to Figure 4-3 but are fit with equation (4.6). Left, high simulation exchange rate of 0.1 s-1_{; right,}

low simulation exchange rate of 0.02 s-1_.

The results in figure 4-2 demonstrated that in a closed system, sequence parameters have a limited effect on the accuracy of exchange rate measurements, and only become a significant source of error at extreme TR, excitation angles, or lower limits of chemical exchange. The closed system model best represents a phantom environment, but it does not realistically model all the characteristics of biological systems. In a perfused system (Figure 4-3), which applies to in vivo studies, sequence

parameters can more significantly impact the measured exchange rates. As shown in figure 4-4 and 4-5, errors are unlikely to be a result of poor SNR except under relatively extreme conditions of very low excitation angles where the low total SNR does correspond to a region of inaccurate exchange rate fitting. The quality of the fit is also not a primary source of these errors. If poor fit quality were the dominate cause of inaccurate fitting results, correlation between higher fit residuals and error in kpl

would be expected. However, as shown in Figure 4-6, fit residuals are either uniform or do not correspond with sequence combinations that result in large kpl errors in the fit (Figures 4-2 and 4-3).